THE MAN, NOT THE METHOD class. 'A mathematician is one to whom that is as obvious as that twice two makes four is to you. Liouville was a mathematician.* Young Hermite's pioneering work in Abelian functions, well begun before he was twenty-one, was as far beyond Kelvin's example in unobviousness as the example is beyond 'twice two makes four.' Remembering the cordial welcome the aged Legendre had accorded the revolutionary work of the young and unknown Jacobi, Liouville guessed that Jacobi would show a similar generosity to the beginning Hermite. He was not mistaken. The first of Hermite1 s astonishing letters to Jacobi is dated from Paris, January 1843. 'The study of your [Jacobf s] memoir on quadruple periodic functions arising in the theory of Abelian functions has led me to a theorem, for the division of the argu- ments [variables] of these functions, analogous to that which you gave ... to obtain the simplest expression for the roots of the equations treated by Abel. M. Liouville induced me to write to you, to submit this work to you; dare I hope, Sir, that you will be pleased to welcome it with all the indulgence it needs?' With that he plunges at once into the mathematics. To recall briefly the bare nature of the problem in question: the trigonometric functions are functions of one variable with one period, thus sin (x + %TT) = sin #, where x is the variable and 2?r is the period; Abel and Jacobi, by 'inverting* the elliptic integrals, had discovered functions of one variable and feoo periods, say f(x -j- p -f q) = /(#), where p, q are the periods (see Chapters 12, 18); Jacobi had discovered functions of fceo variables and four periods, say F(x + a + b, y + c + d) = F(xsy), where ajbtc,d are the periods. A problem early encountered hi trigonometry is to express sin (- ), or sin { -}, or generally sin W W ( -), where n is any given integer, in terms of sin a; (and W possibly other trigonometric functions of or). The correspond- ing problem for the functions of two variables and four periods was that which Hermite attacked. In the trigonometric pro- 499